Nowhere-Zero Unoriented 6-Flows on Certain Triangular Graphs

A nowhere-zero unoriented flow of graph G is an assignment of non-zero real numbers to the edges of G such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A nowhere-zero unoriented k-flow is a flow with values from the set {+/- 1, . . ., +/-(k - 1)}, for short we call it NZ-unoriented k-flow. Let H-1 and H-2 be two graphs, H-1 circle plus H-2 denote the 2-sum of H-1 and H-2, if E(H-1 circle plus H-2) = E(H-1) ? E(H-2), |V (H-1)boolean AND V (H-2)| = 2, and |E(H-1)boolean AND E(H-2)| = 1. A triangle-path in a graph G is a sequence of distinct triangles T-1, T-2, . . ., T-m in G such that for 1 <= i <= m, |E(T-i) boolean AND E(T-i(+1))| = 1 and E(T-i) boolean AND E(T-j) = null if j > i + 1. A triangle-star is a graph with triangles such that each triangle having one common edges with other triangles. Let G be a graph which can be partitioned into some triangle-paths or wheels H-1, H-2, . . ., H-t such that G = H-1 circle plus H-2 circle plus circle plus H-t. In this paper, we prove that G except a triangle-star admits an NZ-unoriented 6-flow. Moreover, if each H-i is a triangle-path, then G except a triangle-star admits an NZ-unoriented 5-flow.